Nuclear Physics with Electromagnetic Probes

Transcription

1 Nuclear Physics with Electromagnetic Probes Lawrence Weinstein Old Dominion University Norfolk, VA Lecture 2 Hampton University Graduate School 2012

2 Course Outline Introduction to EM probes. Electron and photon interactions with nuclei. Beams and detectors. Nuclear Physics with electrons and photons. Elastic electron-nucleus scattering. Charge, matter and momentum distributions. Single nucleons Correlated nucleon pairs. Nucleon modification in nuclei Hadronization in nuclei Nuclear transparency 2

3 Why use electrons and photons? Probe structure understood (point particles) Electromagnetic interaction understood (QED) Interaction is weak (α = 1/137) Perturbation theory works! First Born Approx / one photon exchange Probe interacts only once Study the entire nuclear volume BUT: Cross sections are small Electrons radiate 3

4 It s all photons! An electron interacts with a nucleus by exchanging a single* virtual photon. Scattered e - Real photon Incident e - Virtual photon Real photon: Momentum q = energy ν Mass = Q 2 = q 2 - ν 2 = 0 Virtual photon: Momentum q > energy ν Q 2 = - q μ q μ = q 2 - ν 2 > 0 Virtual photon has mass! (ν and ω are both used for energy transfer) 4

5 (e,e ) spectrum d! d" d! d" Generic Electron Scattering at fixed momentum transfer 5

6 Experimental goals: Elastic scattering structure of the nucleus Form factors, charge distributions, spin dependent FF Quasielastic (QE) scattering Shell structure Momentum distributions Occupancies Short Range Correlated nucleon pairs Nuclear transparency and color transparency Deep Inelastic Scattering (DIS) The EMC Effect and Nucleon modification in nuclei Quark hadronization in nuclei 6

7 Program central to all of nuclear science 7

8 Energy vs length Select spatial resolution and excitation energy independently Photon energy ν determines excitation energy Photon momentum q determines spatial resolution:!!! q Three cases: Low q Photon wavelength λ larger than the nucleon size (R p ) Medium q: 0.2 < q < 1 GeV/c λ ~ R p Nucleons resolvable High q: q > 1 GeV/c λ < R p Nucleon structure resolvable 8

9 Types of photon Polarisation Both real and virtual photons can have polarisation Real photons: transverse pol only Virtual photons: longitudinal or transverse Determining azimuthal distribution of reaction products around these polarisation directions gives powerful information (see later) Linear polarisation (Electric field vector oscillates in a plane) Virtual photon Longitudinal polarisation Circular polarisation (Electric field rotates clockwise or anticlockwise) Virtual photons can also have a longitudinal polarisation component - related to coulomb (charge) scattering 9

10 Real and virtual photon coupling Photon-nucleus EM interaction: Photon EM field 4-vector potential couples to the nuclear charge and current J µ A µ ρφ (e/m)(pxa) x(qxa) Coulomb Convection spin Longitudinal Virtual photon only Transverse Real and Virtual Photons 10

11 Inclusive electron scattering (e,e ) Lab frame kinematics (not detected) Invariants: 11

12 Formalism Inclusive scattering: measure scattering angle θ e and energy E e (ω = E e E e ) and the cross section d! / d" e d# One photon exchange: L μν and H μν are the lepton and hadron tensors 12

13 More Formalism Lepton tensor known (QED): Spin dependent Hadron tensor unknown: Nuclear current operators Two approaches: 1. Calculate J μ from models or 2. Create most general hadron tensor from tensors (bilinear Lorentz vectors) A i μν = a μ b ν and Lorentz scalers S i = a μ b μ : H μν = ΣA i μν Fi (S 1,S 2, ) 13

14 Most general hadron tensor: EM current conservation parity conservation Time reversal symmetry Available independent four vectors for (e,e ): target momentum p μ photon momentum q μ target polarization S μ Four independent structure functions Spin independent: F 1 (Q 2,x) and F 2 (Q 2,x) Spin dependent: g 1 (Q 2,x) and g 2 (Q 2,x) (arguments also written as (Q 2,ω) or (Q 2,ν)) x = Q2 2 p!q = Q2 2M" Spin dependent 14

15 Elastic cross section (p 2 = m 2 ) Recoil factor Form factors Mott cross section F 1, F 2 : Dirac and Pauli form factors G E, G M : Sachs form factors (electric and magnetic) G E (Q 2 ) = F 1 (Q 2 ) - τκf 2 (Q 2 ) G M (Q 2 ) = F 1 (Q 2 ) + κf 2 (Q 2 ) R L, R T : Longitudinal and transverse response fn τ = Q 2 /4M 2 κ = anomalous magnetic moment 15

16 Electron-nucleus interactions I II III 16

17 Electrons as Waves Scattering process is quantum mechanical De broglie wavelength:! = h p Electron energy: E e! pc λ resolving scale :! = 2" (197 MeV#fm) E e 17

18 Analogy between elastic electron scattering and diffraction θ k!! k! q! 18

19 Simple analogy for elastic electron scattering. Classical Fraunhofer Diffraction Amplitude of wave at screen: a 2" % $ # # 0 0 exp ( ibr cos! ) rd! dr 19

20 Classical Fraunhofer Diffraction Intensity: ( 2 ' & $ % J 1 ( 2+a / *) sin) 2 sin) #! " Minima occur at zeroes of Bessel function. 1 st zero: x = some algebra Hence 2 a # 1.22" sin! 20

21 Excursion: Babinet s principle Screen with apertures Patterns appear the same Complementary screen 21

22 Example: 30 Si(e,e ) 1 st minimum = 1.3 fm -1 θ = 32.8 o σ/σ mott q eff (fm -1 ) Electron energy = MeV λ = 2.73 fm Calculated radius = 3.07 fm Measured rms radius = 3.19 fm (from fit to entire curve) (1 fm -1 = 197 MeV/c) 22

23 I. Elastic Electron Scattering from Nuclei (done formally) Fermi s Golden Rule Plane wave approximation for incoming and outgoing electrons Born approximation (interact only once) 23

24 I. Elastic Electron Scattering from (spin-0) Nuclei Form Factor and Charge Distribution Using Coulomb potential from a charge distribution, ρ(x), Charge form factor F(q) is the Fourrier transform of the charge distribution ρ(x) 24

25 I. Elastic Electron Scattering from (spin-0) Nuclei 1 fm -1 = 197 MeV/c σ/σ mott F=1 for point particle q eff (fm -1 ) 25

26 I. Elastic (e,e ) Scattering charge distributions Pb Cross Section Charge density 12 C 40 Ca 4 He 1 q (fm -1 ) Radius (fm) Elastic electron scattering measured for many nuclei over a wide range of Q2(mainly at Saclay in the 1970s) Measured charge distributions agree well with mean field theory calculations. 26

27 Lead 208 Pb Radius Experiment : PREX Pb(e,e ) Elastic Scattering Parity Violating Asymmetry E beam = 1 GeV θ e = 5 o I beam = 60 μa Spokespersons Paul Souder Krishna Kumar Guido Urciuoli Robert Michaels 208 Pb Slides courtesy of R. Michaels 27

28 Parity Violating Asymmetry A PV = σ σ R R σ + σ L L ~ σ e + 0 γ Z e Applications of PV at Jefferson Lab Nucleon Structure (strangeness) -- HAPPEX / G0 Standard Model Tests ( 2 sin ) -- e.g. Qweak Nuclear Structure (neutron density) : PREX θ W

29 A = Lead 208 Pb Radius Experiment : PREX dσ dω dσ dω da A = R R Z 0 of Weak Interaction: Clean Probe Couples Mainly to Neutrons Left-Right Cross section asymmetry (simplified): + dσ dω dσ dω L L = w/ Coulomb distortions: drn 3 % = R n 2 G FQ 1 4sin 2πα 2 1% R n = neutron matter radius 0 2 θ W F W (Q 2 ) F P ( Q 2 ) F W (Q 2 ): 208 Pb Weak Form Factor F p (Q 2 ): 208 Pb Charge Form Factor Clean Probe Couples Mainly to Neutrons proton Electric charge 1 0 neutron (T.W. Donnelly, J. Dubach, I Sick) (C. Horowitz) Weak charge

30 PREX & Neutron Stars ( C.J. Horowitz, J. Piekarewicz ) R n calibrates EOS of Neutron Rich Matter Crust Thickness Explain Glitches in Pulsar Frequency? Combine PREX R n with Obs. Neutron Star Radii Phase Transition to Exotic Core? Strange star? Quark Star? Some Neutron Stars seem too Cold Crab Pulsar Cooling by neutrino emission (URCA) R n R p > 0.2 fm URCA probable, else not

31 PREX in Hall A at JLab Spectrometers Lead Foil Target Hall A Pol. Source CEBAF 31

32 High Resolution Spectrometers Spectrometer Concept: Resolve Elastic Pb 1 st excited state at 2.6 MeV Elastic detector Inelastic Quad target Dipole Q Q 32

33 Lead Target melts easily 208 Pb Liquid Helium Coolant 12 C beam Diamond Backing: High Thermal Conductivity Negligible Systematics Beam, rastered 4 x 4 mm 33

34 Target Assembly 34

35 PRex Results APV = 656 ± 60 (stat) ± 14 (syst) ppb 0.8 R n =R p A Pb PV (ppm) SI NL3p06 SLY4 FSU SIII NL3 PREX-I PREX-II NL3m R n (fm) R p Look for PRex II, coming soon to a lab near you! 35

36 Deuteron Elastic Scattering Only bound 2 nucleon system Prime testing ground for models: where is the limit for the description in terms of nucleons and mesons Questions: How does the photon interact with a BOUND nucleon? Effects of meson interactions? Final State Interactions? Deuteron wave function at small distance? Where do quarks come in? Review Articles: M.Garcon and J.W. van Orden Adv.Nucl.Phys.26(2001)293 R. Gilman and F. Gross, J. Phys. G: Nucl. Part. Phys. 28 (2002) R37 R116 R.J.Holt and R. Gilman 6/11/12 Deuteron slides from W. Boeglin (Florida International U) 36

37 dσ dω = σ Mott Elastic D(e,e ) A(Q 2 )+B(Q 2 ) tan 2 (θ/2) A = G 2 C ηg2 M η2 G 2 Q B = 4 3 η(1 + η)g2 M T 20 = 8 G C Charge form factor G M Magnetic form factor G Q Quadrupole form factor 1 2 +(1+η) tan2 (θ/2) 9 η2 G 2 Q ηg CG Q ηg2 M 2 A + B tan 2 (θ/2). η = Q 2 /4M 2 D Note: A and B are linear combinations of F 1 and F 2 or of R L and R T. To measure the tensor polarization T20, use:! d(e,e')d d(e,e'! d) 6/11/12 37

38 Hall C data A A d Hall A data Photon couples to nucleon d π Meson Exchange Current ~ Tt NRIA G C = Q 2 (GeV/c) 2 Q 2 Calculation: H.Arenhövel et al. PRC 61, B B NRIA + RC +!"# NRIA + RC Hall C data Experiments: Hall A : L.C.Alexa et al. PRL 82 (1999) 1374 Hall C: D.Abbott et al. PRL 82 (1999) 1379 Hall C: D.Abbott et al. PRL 84 (2000) /11/12 38 ρ π ρπγ Meson Exchange Current

39 A A Rel. Calculations in Hamiltonian dynamics: IMII and IM+EII Y.Huang and W.N.Polyzou PRC80 (2009) IMII IM+E II # "! f/g=0 # "! f/g=6.1 No MEC Buchanan Elias Benaksas Arnold Platchkov Galster Cramer Simon Abbott Alexa Berard Akimov Q (GeV ) no MEC contributions T 20 0 T IMII IM+E II # "! f/g=0 # "! f/g=6.1 B Q (GeV ) 6/11/12 Q (fm ) Dmitriev Voitsekhovskii Ferro-Luzzi Schulze Bowhuis -1 Gilman Boden Garcon Abbott Nikolenko Zhang MEC B IMII Q 2 (GeV 2 )d IM+E II # "! f/g=0 # "! f/g=6.1 with MEC Bosted Cramer Auffret Simon Buchanan Rel. Calculations in propagator dynamics: Need ρπγ diagram not well constrained D.R.Phillips et al. PRC72 (2005) )

40 Deuteron Elastic Scattering Summary: Non-Relativistic models fail at the highest Q 2 Relativistic models successfully describe Deuteron form factors MEC contributions are very important ρπγ exchange current important and not well constrained d d π ρ π Photon couples to nucleon Meson Exchange Current ρπγ Meson Exchange Current 6/11/12 40

41 Elastic Scattering Summary Beautiful measurements of the nuclear charge distribution And first measurements of the nuclear matter radius Beautiful measurements of the deuteron A, B and Tensor polarization Allows separation of charge, magnetic and quadrupole form factors m=0 m=±1 41